Does anyone know the name of this curve?

Question

I have come upon the curve with the following parametric equations:

$$x(t)=\log(2+2\cos(t))/2$$ $$y(t)=t/2$$

for $-\pi<t<\pi$. It gives the image in the complex plane under $\log(1+z)$ of the unit circle. Does anyone know whether it has a name? It seems like this must have been studied at some point before.

Greg

Answer 1

This is more or less a reflected and shifted version of the so-called "catenary of equal resistance" (en français, sorry). Here is the paper where they were first studied.

Wikipedia gives a derivation for the equation of the catenary of equal resistance; in some references, this is also called the "catenary of uniform strength". See this for instance.